Getting a bit off topic, but I hope that people don't mind a few factual
>Neither the fifth nor the third is exact in equal
>tempering, but the third is closer, and modern western music uses more
>thirds than fifths.
Nope. The tempered fifth is almost exactly two cents away from the just
fifth (3:2). The tempered major third is about 14 cents away from the
Ptolemaic just major third (5:4) and about eight cents from the Pythagorean
just major third (81:64). Note that the Ptolemaic third is much more
"pure", due to its simpler ratio. (For those not familiar with pitch
measurements, 14 cents is a lot in this context.)
Equal temperament was developed not to prioritize thirds over fifths, but
to allow free modulation between keys.
The chronology goes something like this:
Medieval period: Pythagorean tuning--pure fifths and fourths, pure but
rough thirds, limited ability to modulate.
Renaissance: mean-tone tuning--pure thirds, tempered fifths, more ability
to modulate. Lutenists begin to use equal temperament, which is much easier
to deal with on their instrument. Flame wars between lutenists and
Baroque: Various "well temperaments" compromise both thirds and fifths, but
allow much more modulation. For the first time a keyboard can play
tolerably in all keys, although each has its own unique quality. "The
Well-Tempered Clavier" shows off this capability.
Romantic: Equal temperament becomes the norm. Modulation is all the rage.
All keys theoretically equivalent.
Twentieth century: Equal temperament still going strong, but other forms of
tuning are invented and rediscovered: quarter tones, Ptolemaic intonation,
Partch's Monophony, Lucy Tuning, blue notes, etc., etc.
Note that in all periods, only keyboard instruments strictly obey the
prevailing temperament (and even they cheat a bit for various practical
reasons). Instruments that can vary their pitch continuously tend to do so,
providing what one might describe as "contextual just intonation"--although
the style of this is strongly affected by the prevailing paradigm.
Well, that's a lot of information, of somewhat limited relevance (although
there are probably some interesting typographic parallels to these
developments). I bet you're all glad I bit my tongue during the whole
>it seems to me that, although
>"harmony" can be defined mathematically, our reactions to it are strongly
Tim Walters : The Doubtful Palace : http://www.doubtfulpalace.com